genetic algorithm approach to short-term hydro-thermal scheduling with multiple thermal plants

Electrical Power and Energy Systems 23 (2001) 565±575

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Hybrid simulated annealing/genetic algorithm approach to short-term hydro-thermal scheduling with multiple thermal plants Suzannah Yin Wa Wong Department of Computer and Mathemetics, School of Continuing Studies, Chinese University of Hong Kong, 67 Chatham Road South, 13/F, Kowloon, Hong Kong, People's Republic of China Received 13 August 1998; revised 7 March 2000; accepted 17 April 2000

Abstract This paper ®rst develops a new short-term hydro-thermal scheduling formulation which takes into consideration of scheduling the thermal units as well as the hydro and thermal generations in a schedule horizon consisting of a number of intervals. Ef®cient methods, which can avoid unnecessary commitment or decommitment of thermal generators and the employment of hydro generators to regulate the peak system demands, are developed and incorporated into the formulation to reduce the fuel cost. The formulation is then combined with simulated annealing (SA), genetic algorithms (GAs), and two hybrid optimisation techniques to established SA-based, GA-based and hybrid-based algorithms. The equality constraints, inequality and physical constraints of the hydrothermal systems are fully taken into account in the algorithms. Their performance is validated by applying the algorithms to different test systems. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Short-term hydro-thermal scheduling; Hydro/thermal power systems; Algorithms

1. Introduction In tackling the short-term hydro-thermal scheduling problem, previous approaches, such as Lagrange multiplier techniques [1,2] and the dynamic programming method [3,4], combine the thermal generators into a composite unit. The composite generation production cost curve obtained, hence, is only an approximation of all the input±output curves and may result in loss of accuracy. These techniques, moreover, can only deal with thermal generators, the input±output characteristics of which are approximated by quadratic functions or piece-wise quadratic functions only. In the case that the input±output curves of the generators are highly non-linear and contains discontinuity due to the effects of `valve points' [5], these methods will have dif®culties in determining the composite cost curve. Additionally, in these optimisation techniques, while decisions to commit or de-commit individual thermal unit to meet the varying system load demand cannot be incorporated in the scheduling process, the amount of spinning reserve required and some important operational aspects such as ramp rates of units are not usually included. Moreover, conventional methods based on Lagrange multiplier techniques [1,2] require special measures in their solution processes and they have dif®culties in deter-

mining the solution schedules. The method of dynamic programming [3,4] also has the problem of dimensionality explosion. Hence, in solving the short-term hydro-thermal scheduling problem with multiple thermal plants in the power systems, in order to minimise the total production cost and to obtain better accuracy, the on/off status of each of the thermal plant, as well as the amount of hydro and thermal generations which are to meet the load demands in the schedule horizon, have to be determined. The schedule solution sought by the solution process should be a feasible one which satis®ed all the constraints. In addition, the solution method should be general and the execution of the solution process should not suffer from memory contention. This paper ®rst develops a new short-term hydro-thermal scheduling formulation which takes into consideration of scheduling the thermal units as well as the hydro and thermal generations in a schedule horizon consisting of a number of intervals. Ef®cient methods, which can avoid unnecessary commitment or decommitment of thermal generators and the employment of hydro generators to regulate the peak system demands, are developed and incorporated into the formulation to reduce the fuel cost. The formulation is then combined with simulated annealing (SA) [6,7], genetic algorithms (GAs) [8,9], and two

0142-0615/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0142-061 5(00)00029-6

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S. Yin Wa Wong / Electrical Power and Energy Systems 23 (2001) 565±575

Nomenclature Dj DF F ft(Ptj)) FT I k nj PhjI Phji Poji Pt.max Pt.min Ps t s q(j.max)i q(j.min)i qji r R sji T T0 V(j21)i V(j.max)i V(j.min)i V0i g

total load demand at interval j the change in cost total production cost fuel cost of the generator t operating at power level Ptj at interval j total fuel cost the total number of hydro plants the iteration counter length of interval j the last hydro generation the amount of hydro generation produced by hydro plant i the electric loss from the ith hydro plant to the load in the jth interval maximum operation limits of thermal generator t minimum operation limits of thermal generator t loading of thermal generator t at dependent interval the upper limits of the discharge rate in interval j for reservoir i the lower limits of the discharge rate in interval j for reservoir i water discharge rate of reservoir i at interval j the temperature reduction factor the water travel time the water in¯ow rate and the spillage rate, respectively in interval j for reservoir i temperature initial temperature the volume of reservoir i at interval j the upper volume limits in interval j for reservoir i the lower volume limits in interval j for reservoir i the speci®ed initial volume for reservoir i scaling factor

hybrid optimisation techniques [10±13] to established SA-based, GA-based and hybrid-based algorithms. The equality constraints, inequality and physical constraints of the hydro-thermal systems are taken into account fully in the algorithms. Their performance is validated by applying the algorithms to different test systems. 2. Short-term hydro-thermal scheduling problem with multiple thermal plants The objectives of the short-term scheduling of multiplehydro/thermal plants over a schedule horizon to meet the load demands are: (i) to determine the portion of hydro and thermal generations; (ii) to seek for the on/off schedule of the thermal plants in order to cover the thermal generations; (iii) to set the share of generations by each of the hydro plants and the thermal plants; (iv) the schedule given collectively by items (i), (ii) and (iii) is the most economical in terms of the sum of the start-up costs and the running fuel costs of the thermal generators.

A mathematical description of this scheduling problem is as follows: Assuming there is no cost involved in hydro generations, the objective function, F, to be minimised is given by the total cost of the schedule which consists of the total start-up cost, f; and the total fuel cost, FT, of the thermal generators as in: F ˆ f 1 FT

…1†

For a total of T thermal generators and for J intervals in the schedule horizon, FT is given by FT ˆ

T X tˆ1

‰n1 ft …Pt1 † 1 n2 ft …Pt2 †

1¼ 1 nj ft …Ptj † 1 ¼ 1 nJ ft …PtJ †Š

…2†

where ft(Ptj) is the fuel cost of the thermal generator t operating at power level Ptj in the jth interval in the schedule horizon. ft(Ptj) is set to 0 when generator t is off-lined. The duration of the jth interval is nj. Let there are I variable-head hydro plants each powered by the water of one reservoir. In any interval j, the generation of the ith hydro plant powered by the ith reservoir is Phji ˆ g…V… j21†i ; qji † for i ˆ 1; 2; ¼; I

…3†

S. Yin Wa Wong / Electrical Power and Energy Systems 23 (2001) 565±575

where qji is the water discharge rate and V( j21)i is the volume of reservoir i at the end of the previous interval ( j 2 1). If Dj is the load demand in the jth interval, it is met by the combined hydro and thermal generations according to the following expression: Dj ˆ

I X iˆ1

Phji 1

T X tˆ1

Ptj 2

I X iˆ1

Poji

…4†

in which Poji is the electric loss from the ith hydro plant to the load in the jth interval and it is a function of Phji. The volume of water stored in the ith reservoir at the end of the jth interval, Vji, is governed by the following hydraulic continuity equation: Vji ˆ V… j21†i 1 nj …rji 1 q… j2R†…i21† 2 qji 2 sji † for j ˆ 1; 2; ¼J and i ˆ 1; 2; ¼I

…5†

where qji is the water discharge rate of reservoir i at interval j; q( j2R)(i21), the water discharge rate of reservoir (i 2 1) at ( j 2 R)th interval with R being the water travel time from reservoir (i 2 1) to i; rji and sji, the water in¯ow rate and the spillage rate, respectively in interval j for the reservoir i. In the minimisation of the objective function F in Eq. (1), the reservoir constraints and the operation limits of the hydro plants must be satis®ed in addition to satisfying the power balance requirement in Eq. (4). For interval j, the constraints are summarised below: (a) ith reservoir volume constraints V… j:min†i # Vji # V… j:max†i

…6†

(b) limits of water discharge rate from ith reservoir q… j:min†i # qji # q… j:max†i

…7†

(c) operation limits of ith hydro plant Phji:min # Phji # Phji:max

…8†

Moreover, in the determination of the on/off status of the thermal generators in the minimisation process, all the constraints in the usual unit commitment problem [12] must be included. These constraints are: (a) operation limits of the thermal generators; (b) minimum up and down time constraints; (c) crew constraints; and (d) station shut-down interval constraints. 3. Forming a candidate schedule solution The mixed hydro/thermal generator scheduling problem can be solved using the following approach: (1) Determine the amount of total hydro generation in each schedule interval.

567

(2) From Eq. (4), form the amount of load demand in each interval that is to be met by the thermal generations. (3) Determine the commitment of the thermal generators in the schedule horizon against the load demands obtained in item (2). (4) Items (1)±(3) collectively provide a candidate schedule solution. Combine the method represented by items (1)±(3) with a general optimisation technique to form an algorithm to seek for the global optimum schedule solutions. While the action in item (2) is self-explanatory, those in items (1) and (3) are described in Sections 3.1, 3.2 and 3.3, respectively. The formation of opimisation algorithms in item (4) is presented in Section 4. 3.1. Pro®cient method for setting total hydro generations in schedule intervals In item (1) of the approach, for each of the 1st to the (J 2 1)th interval in a J-interval schedule horizon, the total hydro generation can be determined by the methods developed in Sections 3.1.1 and 3.1.2. 3.1.1. Eliminate unnecessary commitment or decommitment of thermal generators If the total thermal generations in the schedule horizon are almost constant or with little variation, the frequency of the commitment or decommitment of thermal generators can be reduced. Hence, while the start-up cost can be minimised; the ramp rate constraints and the other operative constraints will also be covered with more ease. This can be achieved by the following means: Let Phj and Ph…j11† be the total hydro generations in intervals j and ( j 1 1), respectively. If Ph( j11) can be set to the sum of Phj and the increment or decrement of the load demands between the two intervals, the total thermal generation in these two intervals will then be identical. The rise or the fall of the load demand is here met by the change of the total generation level of the hydro generators. This way offers the advantage of avoiding any unnecessary commitment or decommitment of thermal generators. However, attributed to the capacity of the hydro plants, constant total thermal generations cannot be achieved always. The extent of variation of load demand is reduced by setting Ph( j11) to the sum of Phj and a random amount generated within 0 and the difference between the load demands between intervals j and ( j 1 1). 3.1.2. Regulate peak system demands by hydro plants Unless the hydro capacity is much greater than the thermal capacity, hydro generators are normally used to regulate the peak system demands. They are not employed in the minimum load-demand period, which is normally covered by the base-load generations. Although this can be taken as a guideline to assist the optimisation process to ®nd the

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optimum solution schedule, it should not be strictly followed, otherwise, the optimisation process will be led to a local optimum point. To implement this guideline, the total hydro generations are set to zero in a probabilistic manner in non-peak interval. During the minimum demand period, a random binary number is generated. If it is 0, the total hydro generation is set to zero. Otherwise, the total hydro generation is set by the method in the last section. The method for setting of the total hydro generation in the last interval J and the setting of the generation of each of the hydro generator in all intervals is developed in the next section. 3.2. Setting the generation levels of hydro generators With the total hydro generation in interval 1 to J 2 1 set by the method in Section 3.1, the generation levels of the remaining hydro generators, except the last hydro generator I, are here set by the random numbers generated within the operation ranges of the generators. The loading of the last hydro generator I in each interval is then given by the difference between the total hydro generation and the sum of the hydro generations of the remaining hydro generators in that interval. With the loadings of each hydro generator known in intervals 1 to (J 2 1) and with the speci®ed initial volumes of the reservoirs, from Eqs. (3) and (5), the discharge rate and the water volumes of each reservoir at the beginning and end of intervals 1 to (J 2 1), and at the beginning of interval J can be calculated. Given the ®nal volume of water in reservoir i, for i ˆ 1; 2; ¼; I; at the end of interval J, from the hydraulic continuity equation in expression (5), the water discharge rate in the last interval J can be calculated. The generation of each hydro generator I at the last interval, hence, can be evaluated by Eq. (3). The total hydro generation in the last interval J is then given by the sum of the hydro generations evaluated. 3.3. Determination of the on/off status and generation levels of the thermal units In item (2), the load demands over the schedule horizon to be met by thermal generations are formed. The sub-problem of scheduling thermal generators to meet the load demands obtained in item (2) is solved in item (3). In the present work, the new unit commitment algorithms developed by the author [12] are used to determine the thermal generator schedules and the total costs of the schedules. While the total cost consists of the start-up costs and the operation fuel costs of the thermal generators, the total fuel cost is evaluated by part of the unit commitment algorithms which performs the economic dispatch calculation [11,12]. The new unit commitment algorithms and the new economic dispatch algorithms developed by the authors are based on simulated annealing (SA) [6,7], basic genetic algorithm (BGA) [8], incremental genetic algorithm (IGA)

[9] and the hybrid genetic/simulated annealing algorithms, GAA and GAA2 [10±13]. The pseudo-codes for BGA, IGA, SA and a brief outline of GAA and GAA2 are given in Appendix A. 4. The new algorithms While a method of producing a candidate schedule solution has been developed in Section 3, this method can now be incorporated into the solution process of a general optimisation technique such as SA, BGA, IGA, GAA and GAA2. The following sections describe the formation of new hydro-thermal scheduling algorithms based on these techniques. 4.1. SA-Based hydro-thermal scheduling algorithm With reference to the pseudo-code for SA in Appendix A.1.1, the mechanisms developed in Sections 3.1 and 3.2 are employed in the solution generation process to form the hydro schedules. The scheduling of the thermal generators is solved by the enhanced SA technique developed by the authors [14]. New thermal generator schedule is formed by perturbing the current solution. The perturbation here is to probabilistically delay/advance the on-line/off-line time by one interval. The remaining component of the SA-based hydro-thermal scheduling algorithm are those shown in the pseudo-code. 4.2. Genetic- and hybrid-based hydro-thermal scheduling algorithms When genetic algorithms BGA, IGA and the hybrid algorithms GAA and GAA2 are employed to establish the new hydro-thermal scheduling algorithm, the method for forming a candidate solution in Section 3 is used repeatedly to form the initial solution chromosomes required in the initial population, as shown in the pseudo-codes in Appendices A.1.2±A.1.5 . With reference to these pseudo-codes, subsequent generation of new schedule solutions are produced by the operation of crossover and mutation. The characteristics of the initial schedule solutions that the solutions tend to: (a) avoid unnecessary commitment or decommitment of thermal plants; and (b) to use the hydro generation to level the peak demands should be preserved as much as possible by the crossover and the mutation operation. To achieve this, the chromosome which represents a candidate solution is designed as follows: A candidate solution consists of the loadings of all the thermal and hydro generators, and the operation status of all the thermal plants in the intervals of the schedule horizon. The operation status of the thermal plants are represented by the 2 binary digits and the loadings of the generators are coded using ¯oating-point numbers. Each element in a chromosome is for a schedule interval. Consequently, with a

S. Yin Wa Wong / Electrical Power and Energy Systems 23 (2001) 565±575

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Fig. 1. Representation of schedule solution by A chromosome (Gen t, the tth generator; Int j, the jth schedule interval; Stj, the on/off status of the tth generator in jth interval; Ptj, loading of the tth generator in jth interval; Hyr i, the ith hydro plant. Hij, loading of the ith hydro plant in jth interval).

schedule of J intervals, a chromosome will be make up of J elements. An element is a list consisting of the loadings of the hydro and thermal generators and the on/off status of the thermal generators in that interval. As crossover action is taken at the level of the elements in the chromosomes, the characteristics of the initial schedule solutions will have less chance to be destroyed by crossover. Fig. 1 shows the representation of schedule solution by a chromosome. The objective function of the present scheduling problem is taken to be the total cost function F. In these algorithms, K/F is used as the ®tness function, where K is the maximum ¯oating-point number that can be represented in the computer. It is used to amplify (1/F), the value which is usually small so that the ®tness values of the chromosomes will be in a wider range. Mutation, which can introduce diversity into the population, is achieved by the following way: while the total hydro generation and the outputs of each individual hydro plant are changed by the ways described in Section 3.1 and 3.2, respectively, the mutation mechanisms in the new unit commitment algorithms developed by the author [12], which can maintain the feasibility of the thermal schedule, are employed to set the thermal generator schedules. Crossover is adopted when a random number generated between 0 and 1 is less than or equal to the probability of crossover. Similarly, mutation is applied to a candidate solution when the probability of mutation is greater than or equal to another random number lying between 0 and 1. Table 1 Parameter settings of algorithms Algorithm

BGA, IGA, GAA

GAA2

Population size Number of iterations Probability of crossover Probability of mutation

100 150 0.6 0.001

2 250 1 0.01

5. Application examples The new algorithms developed based on SA, BGA, IGA, GAA and GAA2 in Section 4 have been implemented using the C programming language and tested experimentally. The software systems are run on a PC/586 computer. The new algorithms have been applied to two test examples and the fuel costs are evaluated in Hong Kong currency. Owing to the randomness of these approaches, the algorithms are executed 30 times in the application study. In the following sections, the algorithms are referred to as SA, BGA, IGA, GAA and GAA2. For the two test systems, the settings of the values of the parameters in BGA, IGA and GAA, are given in Table 1 and they are suggested by De Jong [15], except the number of iterations, which is found experimentally. In the table, the population size of 100 means a population of 100 feasible chromosomes. To ensure the speci®ed population size is reached, a maximum of 10,000 crossover operations is allowed in each iteration. For GAA2, the population size and the probability of crossover are 2 and 1, respectively and 60 feasible chromosomes are produced in an iteration. The probability of crossover was set to 1 in order to increase the probability of crossover so as to introduce more changes in the chromosomes. For the execution of the hybrid algorithms GAA2 and GAA, the initial temperature T0 and the temperature reduction rate a in Eq. (A1) are set, respectively to the values of 5000 and 0.98 after some experimentation. In applying the SA-based algorithm to the test system, the temperature reduction factor in Eq. (12) is set to 0.98. The scaling factor g for g.p.d.f. is set to 0.004 and the initial temperature T0 is set to 50,000. The maximum number of iteration is 200 and the number of trials per iteration is 300. All these numbers are determined experimentally. The initial values of the thermal generations over the entire schedule horizon are set randomly.

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S. Yin Wa Wong / Electrical Power and Energy Systems 23 (2001) 565±575

Table 2 Coef®cients of heat-rate functions of generators in test system …F…P† ˆ a p P2 1 b p P 1 c†

Table 3 Generator schedule determined by GAA2 (symbol (2) indicates decommitment)

Gen

Pmax (MW)

Pmin (MW)

a

b

c

a3

a1

c1

b3

2c1

2b3

2a1

2a3

z1 z2 z3 a1 a2 a3 a4 b1 b2 b3 c1 c2 c3

680 360 360 180 180 180 180 180 180 120 120 120 120

0 0 0 60 60 60 60 60 60 40 40 55 55

0.00028 0.00056 0.00056 0.00324 0.00324 0.00324 0.00324 0.00324 0.00324 0.00284 0.00284 0.00284 0.00284

8.1 8.1 8.1 7.74 7.74 7.74 7.74 7.74 7.74 8.6 8.6 8.6 8.6

550 309 307 240 240 240 240 240 240 126 126 126 126

7:15

8:15

8:30

8:45

12:30

17:45

19:30

21:45

5.1. First example The test system is taken from Refs. [1,12]. It consists of a hydro plant and 13 thermal plants, with 3 of them are baseload units. The schedule horizon is 1 day long and there are 96 15-min intervals. The fuel cost functions in dollar per hour of all the thermal units are given in Table 2. The capacities, ramp-rates and generator groups of the generators, as well as other operation limits of the generators can be found in previous work, [11,12] and is not repeated here. The system is shown in Fig. 2. The water discharge rate in acre ft per hour of the hydro plant between 0 and 1000 MW is given in Eq. (9) and that between 1000 and 1100 MW is given in Eq. (10) below. q ˆ 4:97Ph 1 330 q ˆ 0:05…Ph 2 1000†2 1 12…Ph 2 1000† 1 5300

…9† …10†

The electric loss from the hydro plant to the load is discounted. The initial and ®nal volumes of water in the reservoir are 100,000 and 60,000 acre ft, respectively. The minimum and maximum volumes of water are 60,000 and 120,000 acre ft in all intervals. The water in¯ow rate is assumed to be constant at 2000 acre ft/hr and the spillage rate is discounted. For the best hydro-thermal schedule determined by the GAA2 algorithm, while the commitment schedule are

reported in Table 3, the generation outputs are the hydro and thermal plants are summarised in Table 4. Although 96 15-min intervals have been used in the study, loadings of all the generators at each hour are tabulated for clarity. In Table 5, the cost associated with the worst schedules found by each algorithm is tabulated. While the GAA2 can seek for the best solutions in all executions of the algorithms, the worst solution obtained by GAA2 is also of better quality than that of other approaches. This con®rms that the GAA2 algorithm is the most reliable amongst all the methods. Table 6 summarises the execution times of all the algorithms, as reported by previous research before, all the approaches are computationally intensive. 5.2. Second example To further demonstrate the robustness of the GAA2, the software is applied to another test system consisting of two variable-head hydro generators adopted from Ref. [17] and the same thermal plants system from example 1. Fig. 3 is a representation of the system. Moreover, the capability of all the algorithm in seeking for the global or near-global optimum schedule in the presence of valve-point loading in the generator fuel cost characteristics are studied in this example. Coef®cients `e' and `f ' are associated with a sinusoidal component [16], which is superimposed on the quadratic fuel cost function to simulate the valve-point loading [11± 14], are tabulated in Table 7. The reservoir storage and discharge characteristics are summarised in Table 8. The generation of the hydro generator is a function of the water discharge rate and the volume of water in the reservoir. The functions for the ®rst and the second hydro generators in the test system are adopted, respectively from Ref. [17]. Phj1 ˆ 0:001min…qj1 ; 2500†  ‰103 1 3…1 2 exp…20:00001†V… j21†1 †Š

…11†

Phj2 ˆ 0:001min…qj2 ; 2500†  ‰101 1 4…1 2 exp…20:00001†V… j21†2 †Š

Fig. 2. Hydrothermal system in ®rst example at an interval.(H, hydro plant; Ph, loading of hydro plant; B1, B2, B3, base-load thermal units; S1,¼,S10, non-based thermal units; Ps, total loading of the thermal units).

…12†

In Ref. [17], the dispatch penalty factor for the ®rst hydro generator is 1 and that for the second is 1.2 and they are adopted in the present study. The water travel time R from the ®rst to the second reservoir is one hour and the spillage rate is discounted.

Hydro gen (MW)

0 786.60 171.50 281.29 542.71 834.87 923.03 1003.87 1002.37 939.13 980.92 1006.42 1003.14 967.52 928.19 872.39 804.10 701 678.08 557.68 320.78 354.11 289.27 0

Time (h)

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

367.20 11.45 420.01 343.31 675.92 577.32 649.26 610.06 666.51 644.39 593.50 650.76 535.12 514.38 553.57 648.21 457.64 413.41 661.52 574.60 574.60 447.44 365.84 257.29

z1 (MW) 296.27 3.77 175.73 276.12 61.12 260.58 264.48 351.38 321.51 261 304.94 306.19 354.40 359.38 344.40 196.20 244.80 226.80 70.20 83.98 54.49 144.60 263.88 331.11

z2 (MW) 236.53 98.17 212.76 269.28 120.25 222.12 343.23 354.69 309.60 315.48 330.64 296.63 357.33 358.71 303.84 293.20 313.46 308.79 160.20 163.74 200.13 103.85 51.01 341.61

z3 (MW) 0 0 0 0 0 100 60 60 60 60 60 60 60 60 60 60 0 0 0 0 0 0 0 0

a1 (MW) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

a2 (MW) 0 0 0 0 100 60 60 60 60 60 60 60 60 60 60 60 60 60 0 0 0 0 0 0

a3 (MW)

Table 4 Determined hydro-thermal schedule and fuel cost (symbol (1) identi®es time in hour in the next day)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

a4 (MW) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

b1 (MW) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

b2 (MW) 0 0 0 0 0 45.11 40 40 40 40 40 40 40 40 0 0 0 0 0 0 0 0 0 0

b3 (MW) 0 0 0 0 0 20 60 40 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

c1 (MW)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

c2 (MW)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

c3 (MW)

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S. Yin Wa Wong / Electrical Power and Energy Systems 23 (2001) 565±575

Table 5 Costs of the best and worst schedules found by the algorithms

Best Worst

SA

BGA

IGA

GAA

GAA2

2536651.25 2618171.25

2528786.50 2548451.25

2525662.00 2543837.75

2526150.75 2563613.50

2525293.75 2534435.75

The generator schedule and the hydro-thermal schedule of the best solution determined by the GAA2 algorithm are summarised in Tables 9 and 10, respectively. Similar to the ®rst example, loadings of all the generators at hourly interval are tabulated for clarity. In Table 11, the cost associated with the worst schedules found by each algorithm is tabulated. Although GAA2 only seeks for the 2nd best schedule solution, the worst solution it determined, is the cheapest amongst all the methods. Again, it con®rms that the GAA2 algorithm is the most reliable amongst all the algorithms. The best and worst cost of the hydro-thermal schedules determined by the algorithms in 30 executions are tabulated in Table 11 while execution times of the algorithms are summarised in Table 12. From Table 12, it can be seen that GAA2 is quite computational intensive. 6. Conclusions Algorithms based on genetic algorithms, simulated annealing and the combination of these two techniques have been developed for the determination of the optimum short-term hydro-thermal schedules for mixed hydro/thermal power systems. In the algorithms, all the thermal generators are considered individually and they are not represented by an equivalent thermal generator unit. The advantage of this approach is that the characteristics of the thermal generators are re¯ected fully into the algorithms. In particular, valve-point loadings can be explicitly included into the heat-rate curves and be dealt with by the algorithms. A method of forming a candidate schedule solution and an ef®cient technique for setting the level of hydro generations have been developed. The method and the technique enhance the performance of the new algorithms. The application study shows that the hybrid algorithm GAA2 is the most reliable algorithm among all the new algorithms although all of them have the ability to seek for the global optimum schedule solution. The computation speed of GAA2 and the applicability of this algorithm can be increased by means of parallel processing as in Ref. [18].

Acknowledgements The valuable discussion provided by staff of the China Light and Power Company, Victor, Lee Yuk Wah, Hong Kong is gratefully acknowledged. Appendix A A.1. Simulated annealing technique, basic genetic algorithm, incremental genetic algorithm, genetic/ annealing algorithm and genetic/annealing algorithm-2 Pseudo-code describing the SA, BGA, IGA, GAA and GAA2 is as follows: A.1.1. Simulated annealing technique Let the maximum number of trials in an iteration be M and the maximum number of iterations be K. The pseudocode for the simulated annealing technique is Initialise current solution and temperature T0 For iteration k ˆ 1 to K do { For trial m ˆ 1 to M do { Set acceptance ¯ag to 0; / p Solution generation mechanism p / Generate a new solution by perturbing the current solution; Evaluate the change in cost, DE; in the solutions / p Solution acceptance test p / If …DE , 0† { Set acceptance ¯ag to 1; } else { If probability of acceptance Pr…DE† . ‰randŠ; set acceptance ¯ag to 1; }

Table 6 Shortest and longest execution time (in min) by the algorithms Method

Shortest

Longest

Average

SA BGA IGA GAA GAA2

54.56867 8.586 13.450 11.383 71.908

1090.516 11.612 17.375 18.486 97.565

406.3413 10.367 15.777 14.456 79.579

Fig. 3. Hydrothermal system in second example at an interval. H1 and H2, hydro plants; Ph, total loading of hydro plants; B1, B2,. B3, base-load thermal units; S1,¼,S10, non-based thermal units; Ps, total loading of the thermal units.

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Table 7 Valve points loading of generators …F…P† ˆ a p P2 1 b p P 1 c 1 ue sin… f …Pmin 2 P††u† Gen

z1

z2

z3

a1

a2

a3

a4

b1

b2

b3

c1

c2

c3

e f

300 0.035

200 0.042

200 0.042

150 0.063

150 0.063

150 0.063

150 0.063

150 0.063

150 0.063

100 0.084

100 0.084

100 0.084

100 0.084

Table 8 Characteristics of the reservoirs Res. no.

Min. vol (acre ft)

Max. vol (acre ft)

Start vol. (acre ft)

End vol. (acre ft)

Min. discharge (acre ft/hr)

Max. discharge (acre ft/hr)

In¯ow rate (acre ft/hr)

1 2

5000 5000

20 000 19 000

6000 18 000

16 000 8000

0 0

3000 2900

1000 0

}

If accepted, use new solution as the current solution; } / p Decrease temperature p / Reduce temperature level according to Tk11 ˆ r k T0 / p repeat for another iteration p /

Crossover to produce a new generation of P…g 1 1†; Perform mutation on P…g 1 1†; End; A.1.3. Incremental genetic algorithm Initialise P; / p P is a population of size S p / Evaluate ®tness of chromosomes in P; While (not terminate) { Set size counter s to 0; While (s , S) { Select parents for crossover based on their ®tness; Crossover to produce two new children to replace two chromosomes in P; Evaluate ®tness of chromosomes of P; s ˆ s 1 2; } Perform mutation on P; } }

In the pseudo-code above, [rand] is a random number uniformly distributed over the interval 0±1; r, the temperature reduction factor; T0, the initial temperature. The probability of acceptance Pr(DE) is given by [19] Pr…DE† ˆ ‰1=… 1 1 exp…DE=Tk ††Š

…A1†

A.1.2. Basic genetic algorithm Set the generation counter g to 0; Initialise population of chromosomes P(g) at generation 0; Evaluate ®tness of chromosomes in P(g); While (not terminate) { Generate P(g 1 1) from P(g); g ˆ g 1 1; Evaluate ®tness of chromosomes of P(g); }

A.1.4. Genetic/annealing algorithm While the ¯ow of GAA is similar to the IGA, its performance is enhanced by adopting the acceptance test in simulated annealing as the replacement test. Chromosomes in the population are replaced by new chromosomes generated either after the crossover operation or the mutation operation. The replacement test is as follows:

A.1.2.1. Pseudo-code for generating P(g 1 1) from P(g)

Evaluate the change in cost, DE; in the new child and a chromosome in P / p Solution replacement test p /

Begin Select parent chromosomes for crossover on the basis of their ®tness; Table 9 Generator schedule determined by GAA2 On-lined time Unit

5:45 a1

6:00 b3

6:45 a3

7:00 b2

7:45 c2

8:00 a2

8:45 b1

9:00 c3

Off-lined time Unit

16:30 2b3

18:30 2b2

20:15 2c2

20:45 2c3

22:15 2a2

23:15 2a1

24:00 2a3

26:30 2b1

567.80 637.84 598.40 673.55 672.06 680 680 680 680 680 680 680 680 680 680 680 680 668.79 619.26 541.28 468.05 477.65 530.56 393.24

265.43 113.25 169.70 210.96 254.33 360 360 360 360 360 360 360 360 360 360 360 360 350.37 341.28 349.00 318.44 353.60 97.92 207.72

66.77 148.91 165.68 148.90 338.61 360 360 360 360 360 360 360 360 360 360 360 360 358.85 331.90 309.72 303.51 158.75 281.52 329.04

0 0 33.33 60 60 179.16 108.24 75.72 122.16 174.48 139.08 113.52 151.56 122.88 60 70.71 166.44 60 60 60 0 0 0 0

0 0 0 0 0 61.83 105.00 162.24 150.36 99.69 60.60 61.06 109.08 164.88 60.84 122.40 64.32 60 60 0 0 0 0 0

0 0 0 33.33 60 62.87 158.64 102.96 129.00 60.70 60.67 110.76 148.68 63.87 61.32 165.48 66.47 60 60 60 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 250.93 21.46 250.93 251.00 251.05 219.08 251.27 156.07 251.49 215.54 174.27 0 38.64 7.57 0 0 0 0 0

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 11 12 13

0 0 12.88 3.25 0 0 256.76 186.57 79.09 105.05 256.89 220.70 0 0 256.89 0 1.80 59.79 31.24 0 0 0 0 0

2nd hydro (MW) z1 (MW) z2 (MW) z3 (MW) a1 (MW) a2 (MW) a3 (MW) a4 (MW)

Time (hr) 1st hydro (MW)

Table 10 Optimum hydro-thermal schedule determined by GAA2 (symbol (1) identi®es time in hour in the next day)

4.00 4.00 4.00 4.00 4.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 4.00

0 0 0 0 0 33.33 66.12 172.56 179.52 60.01 60.03 66.72 152.40 113.16 60.26 106.63 61.82 60 60 60 60 60 60 0

0 0 0 0 60 77.05 76.98 60.61 63.20 60.51 60.12 83.87 117.48 158.28 60.74 0 0 0 0 0 0 0 0 0

0 0 0 40 40 40.14 65.28 40 46.99 40.15 40.02 40.25 56.84 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 15.00 56.51 85.10 55.11 59.05 55.04 55.02 58.39 78.50 68.79 55.34 64.43 55.56 0 0 0 0 0 0 0

0 0 0 0 0 0 60 55.11 61.46 55.16 55.00 55.33 65.40 58.57 55.00 55.14 63.60 0 0 0 0 0 0 0

b1 (MW) b2 (MW) b3 (MW) c1 (MW) c2 (MW) c3 (MW)

574 S. Yin Wa Wong / Electrical Power and Energy Systems 23 (2001) 565±575

S. Yin Wa Wong / Electrical Power and Energy Systems 23 (2001) 565±575

References

Table 11 Costs of the best and worst hydro-thermal schedules Methods

Best cost

Worst cost

SA BGA IGA GAA GAA2

47528250 4302388.00 4292295.00 4299193.50 4295212.50

48538512 4342262.00 4324748.50 4327439.50 4322957.50

575

If (DE , 0) { Set replacement identi®er to 1; } else { If probability of replacement Pr…DE† . ‰randŠ; set replacement identi®er to 1; } If replaced, replace the chromosome in P; The probability of replacement Pr(DE) is given by the same expression as in Eq. (A1) A.1.5. Genetic/annealing algorithm-2 By restricting the population size of GAA to 2 chromosomes, the memory requirement is minimised and the resultant algorithm is called GAA2. To maintain the diversity of the population, the mutation rate is increased with probability of crossover set to 1. With the same replacement test as GAA, its ¯ow is as follows: Initialise P; / p P is a population of size 2 p / While (not terminate) { Set size counter s to 0; While (s , S) { / p S is a pre-assigned number p / Crossover to produce two new children replacement test to replace two parent chromosomes; s ˆ s 1 2; } Perform mutation on P; if [rand] . ˆ 0.5 replace one of the chromosome by the ®ttest chromosome generated so far } Similar to the SA algorithm, [rand] is a random number uniformly distributed over the interval 0±1.

Table 12 Execution times (in min) of the new algorithms Methods

Shortest

Longest

Average

SA BGA IGA GAA GAA2

310.85 6.57 9.46 10.63 41.87

3395.71 25.43 31.61 50.60 96.55

2028.93 9.35 13.13 16.00 57.84

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